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272                       ANALYTICAL MECHANICS
placement.   Eliminating co between equations (1) and (2) we obtain
v = 1L V2ga (/ + !') (M+m) (1 - cos a}.           (3>
The moment of inertia of the target may be determined by observing the period of oscillation when it is used as a pendulum. It will be shown later* that if P denotes the period then
Eliminating (! + /') between equations (3) and (4) we get Pabg (M+m)   jl — cos
TTjt                         T                  2i
__ Pabg (M+m)   .  2^
r/                     O
TTJL                                         £
But in practice m is very small compared with M, the bullet, is small enough to be considered as a small particle, and a is. small; therefore we can neglect m in the numerator, substitute mb2 for I', and replace sin - by -. When these simplifications are introduced into equation (5) we get
4|    PagM  2                                       ,„.
V= ----s—r- a.                                           (6)
4 irmb
213. Motion Relative to the Center of Mass.—Suppose a rigid body to have a uniplanar motion. Let M be the mass of the body, I its moment of inertia with respect to an axis perpendicular to the plane of the motion, Ic its moment of inertia about a parallel axis through the center of mass, and a the distance between the two axes. Then the angular momentum about the first axis is
where v is the velocity of the center of mass.    In the right hand member of the last equation the first term represents
* Page 309.